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Applications of Similar Triangles

🔄 Quick Recap​

In the previous sections, we've learned about similar triangles and the criteria to determine when two triangles are similar (AA, SSS, and SAS). Now, let's explore how this knowledge can be applied to solve real-world problems.

📚 How Similar Triangles Help Us Measure the World​

Similar triangles are incredibly useful tools for making measurements that would otherwise be difficult or impossible to take directly. Let's see how they help us in various situations.

Applications of Similar Triangles

🌍 Real-Life Applications​

Measuring Heights of Tall Objects​

Have you ever wondered how we can measure the height of very tall structures like trees, buildings, or mountains without actually climbing them? Similar triangles make this possible!

Example: Finding the height of a tree​

Let's say we want to find the height of a tree. We can:

  1. Place a mirror on the ground at point C
  2. Stand at point B where you can see the top of the tree (point A) in the mirror
  3. Measure the distances BC, CD, and your height BE

By the laws of reflection, angle ABC = angle DCE. This means triangles ABC and DCE are similar (by the AA criterion).

Therefore:

AB/DC = BC/CE

From this, we can calculate AB, which is the height of the tree!

Measuring Distances Across Rivers​

Similar triangles can help us measure the width of a river without crossing it.

Example: Finding the width of a river​

  1. Stand at point A on one bank of the river
  2. Mark a point C directly across from you on the other bank
  3. Walk a known distance to point B downstream
  4. From B, sight a point D on the opposite bank such that BD is perpendicular to AB
  5. Measure the distance from B to D

Triangles ABC and ABD are similar (by the AA criterion), so:

AC/BD = AB/AB

Since AB = AB, we have AC = BD, which gives us the width of the river!

🧮 Mathematical Applications​

Pythagoras Theorem Using Similar Triangles​

Did you know that we can prove the famous Pythagoras Theorem using similar triangles? Here's how:

In a right-angled triangle ABC with right angle at B:

  1. Draw an altitude BD from B to AC
  2. This creates three similar triangles: ABC, ABD, and BDC

From the similarity of these triangles, we can derive:

AB² = AC × AD
BC² = AC × DC

Adding these equations:

AB² + BC² = AC × AD + AC × DC = AC × (AD + DC) = AC²

And there we have the Pythagoras Theorem: a² + b² = c²!

Similar Triangles in Coordinate Geometry​

Similar triangles are also useful in coordinate geometry. For example, if we have a line with slope m:

  • Any two right-angled triangles formed by this line with the coordinate axes will be similar
  • This similarity helps us derive the formula for the distance between two points

✅ Solved Examples​

Example 1: Finding the height of a pole​

A girl of height 1.5 meters is standing at some distance from a lamp post. The length of her shadow is 4.5 meters and the length of the shadow of the lamp post is 12 meters. Find the height of the lamp post.

Solution: Let's denote:

  • Height of the girl = 1.5 m
  • Length of girl's shadow = 4.5 m
  • Length of lamp post's shadow = 12 m
  • Height of the lamp post = h (to be found)

The triangles formed by the girl and her shadow, and by the lamp post and its shadow, are similar (by the AA criterion).

Therefore:

Height of girl / Girl's shadow = Height of lamp post / Lamp post's shadow
1.5 / 4.5 = h / 12
1.5 × 12 = 4.5 × h
18 = 4.5h
h = 18 / 4.5 = 4 meters

So, the height of the lamp post is 4 meters.

Example 2: Distance across a river​

A surveyor wants to find the distance across a river. She stands at point A and spots a tree at point C directly across the river. She then walks 100 meters downstream to point B and finds that point C is at an angle of 60° from the river bank. Find the width of the river.

Solution: In the right-angled triangle ABC:

  • AB = 100 m
  • Angle ACB = 90° (since C is directly across from A)
  • Angle CAB = 90° - 60° = 30°

Using trigonometry in the right-angled triangle:

tan(30°) = AC / AB
AC = AB × tan(30°)
AC = 100 × (1/√3)
AC = 100 / √3 = 100 × √3/3 ≈ 57.7 meters

So, the width of the river is approximately 57.7 meters.

🧪 Activity Time!​

Measuring the height of a tall object​

What you'll need:

  • A meter stick or measuring tape
  • A partner
  • A sunny day

Steps:

  1. Have your partner stand in the sun so they cast a shadow
  2. Measure:
    • Your partner's height
    • The length of your partner's shadow
    • The length of the shadow of the tall object (e.g., a tree or pole)
  3. Using the formula based on similar triangles: 
    Height of object = (Object's shadow × Person's height) / Person's shadow
  4. Calculate the height of the tall object
  5. Compare your result with estimates made by others in your class

⚠️ Common Misconceptions​

  • Misconception: Similar triangles can only be used for measuring heights. Truth: Similar triangles can be used for measuring distances, depths, and many other dimensions.

  • Misconception: The shadow method only works on sunny days. Truth: While the shadow method needs light to work, there are other methods based on similar triangles that don't require shadows.

💡 Key Points to Remember​

  • Similar triangles help us measure heights and distances that would be difficult to measure directly.
  • The key to using similar triangles for measurement is to create a situation where you know some measurements and can use corresponding parts of similar triangles to find unknown measurements.
  • Applications include:
    • Measuring heights of tall objects
    • Finding distances across inaccessible areas
    • Proving mathematical theorems like the Pythagoras Theorem
    • Solving problems in physics, engineering, and architecture

🤔 Think About It!​

  1. How might architects use similar triangles when designing buildings?
  2. Can you think of a way to measure the height of your school building using similar triangles?
  3. How might ship captains have used similar triangles for navigation before modern technology?

🔜 What Next?​

Now that you understand similar triangles and their applications, you're ready to:

  • Apply these concepts to more complex problems
  • Learn about other geometric properties and theorems
  • Explore how these concepts are used in various fields like astronomy, engineering, and architecture

In the next section, we'll summarize everything we've learned about triangles in this chapter!